Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

LIMIT THEOREMS FOR A LOCALIZATION MODEL OF 2-STATE QUANTUM WALKS

Takuya Machida

Published Jan 1, 2010• 12 citations
OpenAlex

Abstract

We consider 2-state quantum walks (QWs) on the line, which are defined by two matrices. One of the matrices operates the walk at only half-time. In the usual QWs, localization does not occur at all. However, our walk can be localized around the origin. In this paper, we present two limit theorems, that is, one is a stationary distribution and the other is a convergence theorem in distribution.

Related papers

LIMIT THEOREMS FOR A LOCALIZATION MODEL OF 2-STATE QUANTUM WALKS

TAKUYA MACHIDA

We consider 2-state quantum walks (QWs) on the line, which are defined by two matrices. One of the matrices operates the walk at only half-time. In the usual QWs, localization does not occur at all. However, our walk can be localized around the origin. In this paper, we present two limit theorems, that is, one is a stationary distribution and the other is a convergence theorem in distribution.

arXiv:1008.1116•International Journal of Quantum Information•Apr 1, 2011•6 citations

Limit distribution with a combination of density functions for a 2-state quantum walk

Takuya Machida

We consider 2-state quantum walks (QWs) on the line, which are defined by two matrices. One of the matrices operates the walk in certain intervals. In the usual QWs starting from the origin, localization does not occur at all. However, our walk can be localized around the origin. In this paper, we present some limit distributions for the walk.

arXiv:1010.3481•Oct 18, 2010•3 citations

WEAK LIMITS FOR QUANTUM WALKS ON THE HALF-LINE

CHAOBIN LIU, NELSON PETULANTE

For a discrete two-state quantum walk (QW) on the half-line with a general condition at the boundary, we formulate and prove a weak limit theorem describing the terminal behavior of its transition probabilities. In this context, localization is possible even for a walk predicated on the assumption of homogeneity. For the Hadamard walk on the half-line, the weak limit is shown to be independent of the initial coin state and to exhibit no localization.

arXiv:1212.1109•International Journal of Quantum Information•Sep 1, 2013•7 citations

Limit theorems of a two-phase quantum walk with one defect

Shimpei Endo, Takako Endo, Norio Konno +2 more

We treat a position dependent quantum walk (QW) on the line which we assign two different time-evolution operators to positive and negative parts respectively. We call the model “the two-phase QW” here, which has been expected to be a mathematical model of the topological insulator. We obtain the stationary and time-averaged limit measures related to localization for the two-phase QW with one defect. This is the first result on localization for the two-phase QW. The analytical methods are mainly based on the splitted generating function of the solution for the eigenvalue problem, and the generating function of the weight of the passages of the model. In this paper, we call the methods “the splitted generating function method” and “the generating function method”, respectively. The explicit expression of the stationary measure is asymmetric for the origin, and depends on the initial state and the choice of the parameters of the model. On the other hand, the time-averaged limit measure has a starting point symmetry and localization effect heavily depends on the initial state and the parameters of the model. Regardless of the strong effect of the initial state and the parameters, the time-averaged limit measure also suggests that localization can be always observed for our two-phase QW. Furthermore, our results imply that there is an interesting relation between the stationary and time-averaged limit measures when the parameters of the model have specific periodicities, which suggests that there is a possibility that we can analyze localization of the two-phase QW with one defect from the stationary measure.

arXiv:1409.8134•Quantum Information and Computation•Nov 1, 2015•22 citations

Limit theorems for the discrete-time quantum walk on a graph with joined half lines

Kota Chisaki, Norio Konno, Etsuo Segawa

We consider a discrete-time quantum walk W_{t,\kappa} at time t on a graph with joined half lines J_\kappa, which is composed of \kappa half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with symmetric initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that W_{t,\kappa} can be reduced to the walk on a half line even if the initial state is asymmetric. For W_{t,\kappa}, we obtain two types of limit theorems. The first one is an asymptotic behavior of W_{t,\kappa} which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for W_{t,\kappa}. On each half line, W_{t,\kappa} converges to a density function like the case of the one-dimensional lattice with a scaling order of t. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.

arXiv:1009.1306•Quantum Information and Computation•Mar 1, 2012•12 citations

Limit theorems of a 3-state quantum walk and its application for discrete uniform measures

Takuya Machida

We present two long-time limit theorems of a 3-state quantum walk on the line when the walker starts from the origin. One is a limit measure which is obtained from the probability distribution of the walk at a long-time limit, and the other is a convergence in distribution for the walker’s position in a rescaled space by time. In addition, as an application of the walk, we obtain discrete uniform limit measures from the 3-state walk with a delocalized initial state.

arXiv:1404.1522•Quantum Information and Computation•Apr 1, 2015•25 citations
We consider 2-state quantum walks (QWs) on the line, which are defined by two matrices. One of the matrices operates the walk at only half-time. In the usual QWs, localization does not occur at all. However, our walk can be localized around the origin. In this paper, we present two limit theorems, that is, one is a stationary distribution and the other is a convergence theorem in distribution.
We treat a position dependent quantum walk (QW) on the line which we assign two different time-evolution operators to positive and negative parts respectively. We call the model “the two-phase QW” here, which has been expected to be a mathematical model of the topological insulator. We obtain the stationary and time-averaged limit measures related to localization for the two-phase QW with one defect. This is the first result on localization for the two-phase QW. The analytical methods are mainly based on the splitted generating function of the solution for the eigenvalue problem, and the generating function of the weight of the passages of the model. In this paper, we call the methods “the splitted generating function method” and “the generating function method”, respectively. The explicit expression of the stationary measure is asymmetric for the origin, and depends on the initial state and the choice of the parameters of the model. On the other hand, the time-averaged limit measure has a starting point symmetry and localization effect heavily depends on the initial state and the parameters of the model. Regardless of the strong effect of the initial state and the parameters, the time-averaged limit measure also suggests that localization can be always observed for our two-phase QW. Furthermore, our results imply that there is an interesting relation between the stationary and time-averaged limit measures when the parameters of the model have specific periodicities, which suggests that there is a possibility that we can analyze localization of the two-phase QW with one defect from the stationary measure.
We consider a discrete-time quantum walk W_{t,\kappa} at time t on a graph with joined half lines J_\kappa, which is composed of \kappa half lines with the same origin. Our analysis is based on a reduction of the walk on a half line. The idea plays an important role to analyze the walks on some class of graphs with symmetric initial states. In this paper, we introduce a quantum walk with an enlarged basis and show that W_{t,\kappa} can be reduced to the walk on a half line even if the initial state is asymmetric. For W_{t,\kappa}, we obtain two types of limit theorems. The first one is an asymptotic behavior of W_{t,\kappa} which corresponds to localization. For some conditions, we find that the asymptotic behavior oscillates. The second one is the weak convergence theorem for W_{t,\kappa}. On each half line, W_{t,\kappa} converges to a density function like the case of the one-dimensional lattice with a scaling order of t. The results contain the cases of quantum walks starting from the general initial state on a half line with the general coin and homogeneous trees with the Grover coin.